A matching theorem for graphs

Abstract

Let the vertices of an undirected graph be given labels 1, 2, …, n, 1′, 2′, …, n′ such that each vertex has at least n−1 different labels without both i and i′ for any i. Then among all paths between a vertex labeled i and a vertex labeled i′ for any i, the maximum number which are mutually edge disjoint equals the minimum size of an edge cut-set separating all vertices labeled i from all those labeled i′ for any i.